# EPSG:1056

## Time-dependent Coordinate Frame rotation (geocen)

### Attributes

Data source: EPSG

Information source: EPSG guidance note #7-2, http://www.epsg.org

Revision date: 2019-02-21

Remarks: Note the analogy with the Time-dependent Position Vector transformation (code 1053) but beware of the differences! The Position Vector convention is used by IAG. See method codes 1057 and 1058 for similar methods operating between other CRS types.

### Formula

```Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.

The transformation between source and target CRS geocentric coordinates is a two-step process. The first step is to compute the values of seven Helmert transformation parameters at the required epoch. The second step is to apply a seven-parameter Helmert transformation to the source CRS coordinates. This Coordinate Frame rotation is expressed in matrix form as:

(Xt)        (  1     +rZ'  -rY')     (Xs)    (tX')
(Yt) = M' * ( -rZ'    1    +rX')  *  (Ys)  + (tY')
(Zt)        ( +rY'  -rX'     1 )     (Zs)    (tZ')

where
tX' = tX + dtX (t – t0)
tY' = tY + dtY (t – t0)
tZ' = tZ + dtZ (t – t0)
rX' = rX + drX (t – t0)
rY' = rY + drY (t – t0)
rZ' = rZ + drZ (t – t0)
dS' = dS + ddS (t – t0)
M' = 1 + dS'

and the parameters are defined as:
(tX', tY', tZ'): Translation vector, to be added to the point's position vector
in the source coordinate reference system in order to transform from source
coordinate reference system to target coordinate reference system; also: the
coordinates of the origin of source coordinate reference system in the target
frame.

(rX', rY', rZ'): Rotations to be applied to the coordinate reference frame.
The sign convention is such that a positive rotation of the frame about an
axis is defined as a clockwise rotation of the coordinate reference frame when
viewed from the origin of the Cartesian coordinate reference system in the positive
direction of that axis, that is a positive rotation about the Z-axis only from
source coordinate reference system to target coordinate reference system will
result in a smaller longitude value for the point in the target coordinate reference
system. Although rotation angles may be quoted in any angular unit of measure,
the formula as given here requires the angles to be provided in radians.

M': The time-adjusted scale correction to be made to the position vector in the source coordinate reference system in order to obtain the correct scale in the target coordinate reference system. M' = (1 + dS'). When the time-adjusted scale difference dS' is expressed in parts per million, M' = (1 + dS'*10^-6). When the time-adjusted scale difference dS' is expressed in parts per billion, M' = (1 + dS'*10^-9).```

### Example

```Transformation from ITRF2008 to GDA94 at epoch 2013.90.

Source CRS ITRF2008 coordinates at epoch 2013.9
(geocentric Cartesian coordinates):
Xs = –3789470.710 m
Ys = 4841770.404 m
Zs = –1690893.952 m
t = 2013.90

Transformation parameter values:

tX = –84.68 mm
dtX = +1.42 mm/yr
tY = –19.42 mm
dtY = +1.34 mm/yr
tZ = +32.01 mm
dtZ = +0.90 mm/yr
rX = –0.4254 msec
drX = +1.5461 msec/yr
rY = +2.2578 msec
drY = +1.1820 msec/yr
rZ = +2.4015 msec
drZ = +1.1551 msec/yr
dS = +0.00971 ppm
ddS = +0.000109 ppm/yr
t0 = 1994.00

First apply the correction due to rate of change to each of the 7 transformation parameters for the period (t-t0), taking care to convert the translations to the same units as the source CRS (in this case metres) and the rotations to radians:
tX' = –84.68 + [+1.42 * (2013.90-1994.00)] = –56.42 mm = –0.056m
tY' = –19.42 + [+1.34 * (2013.90-1994.00)] = +7.25 mm = +0.007 m
tZ' = +32.01 + [+0.90 * (2013.90-1994.00)] = 49.92 mm = +0.050 m
rX' = –0.4254 + [(+1.5461 * (2013.90-1994.00)] = +30.3420 msec =  +1.471021E-07rad
rY' = +2.2578 + [(1.1820 * (2013.90-1994.00)] = +25.7796 msec = +1.249830E-07 rad
rZ' = +2.4015 + [(1.1551 * (2013.90-1994.00)] = +25.3880 msec = +1.230844E-07 rad
dS' = +0.00971 + [+0.000109 * (2013.90-1994.00)] = +0.01188 ppm

Then M' = 1.00000001188

Using these time-adjusted parameter values, application of the 7 parameter Coordinate Frame rotation formula to the given (source) ITRF2008 coordinates results in:
Xt = –3789470.004 m
Yt =  4841770.686 m
Zt = –1690895.108 m
on the GDA94 (target) geocentric coordinate reference system.

For the reverse transformation from GDA94 coordinates to ITRF08 coordinates at epoch 2013.9, the signs of all parameters need to be reversed except for the reference epoch. Then:
tX' = +84.68 + [(–1.42 * (2013.90 –1994.00)] = +56.42 mm = +0.056 m
and similarly for the other six parameters. Hence tY' = –0.007 m, tZ' = –0.050 m, rX' = –1.471021E-07 rad, rY' = –1.249830E-07 rad, rZ' = –1.230844E-07 rad and dS' = –0.01188 ppm.

Using these time-adjusted parameters values, M' = 0.99999998812 and then applying the 7 parameter Coordinate Frame rotation fomula to the GDA94 coordinates of
Xs = –3789470.004 m
Ys = 4841770.686 m
Zs = –1690895.108 m

results in ITRF08 coordinates at epoch 2013.9 of:

Xt = –3789470.710 m
Yt = 4841770.404 m
Zt = –1690893.952 m```